"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

Archive for April, 2010

Today we’re going to relate the representation graphs introduced in this blog post to something I blogged about in the very first and second posts in this blog! The result will be a beautiful connection between the finite subgroups of , the Platonic solids, and the ADE Dynkin diagrams. This connection has been written about in several other places on the internet, for example here, but I don’t know that any of those places have actually gone through the proof of the big theorem below, which I’d like to (as much for myself as for anyone else who is reading this).

Let be a finite subgroup of . Since any inner product on can be averaged to a -invariant inner product, every finite subgroup of is conjugate to a finite subgroup of , so we’ll suppose this without loss of generality. The two-dimensional representation of coming from this description is therefore faithful and self-dual. Consider the representation graph , whose vertices are the irreducible representations of and where the number of edges between and is the multiplicity of in . We will see that is a connected undirected loopless graph whose spectral radius is . Today our goal is to prove the following.

We’ll describe these graphs later; for now, just keep in mind that they are graphs with a number of vertices which is one greater than their subscript. In a later post we’ll see how these give us a classification of the Platonic solids, and we’ll also discuss other connections.

be a graded representation of , i.e. a functor from to the category of graded vector spaces with each piece finite-dimensional. Thus acts on each graded piece individually, each of which is an ordinary finite-dimensional representation. We want to define a character associated to a graded representation, but if a character is to have any hope of uniquely describing a representation it must contain information about the character on every finite-dimensional piece simultaneously. The natural definition here is the graded trace

.

In particular, the graded trace of the identity is the graded dimension or Hilbert series of .

Classically a case of particular interest is when for some fixed representation , since is the symmetric algebra (in particular, commutative ring) of polynomial functions on invariant under . In the nicest cases (for example when is finite), is finitely generated, hence Noetherian, and is a variety which describes the quotient .

In a previous post we discussed instead the case where for some fixed representation , hence is the tensor algebra of functions on . I thought it might be interesting to discuss some generalities about these graded representations, so that’s what we’ll be doing today.

Some people use the Rubik’s cube group to motivate group theory. I’m a fan of hands-on mathematics, and there’s a lot to learn from the cube; for example, you quickly understand that groups are not in general commutative. The Rubik’s cube itself is also a good example of a torsor.

Actually, just so this post has some mathematical content, there’s something about the Rubik’s cube group that is probably very simple to explain, but which I don’t completely understand. It’s a common feature of Rubik’s cube algorithms that they need to switch around some parts of the cube without disturbing others; in other words, the corresponding permutation needs to have a lot of fixed points. This seems to be done by writing down a lot of commutators, but I’m not familiar with any statements in group theory of the form “commutators tend to have fixed points.” Can anyone explain this?